Window Aperture Sizing

Window Aperture Sizing (vehicle-design/sizing/window-aperture-sizing)

Size a pressurized-cabin passenger window aperture as a flat circular pane clamped at its edge under a uniform pressure differential. This leaf computes the design differential from the ISA pressures at the cabin and flight altitudes with the certification pressure factor, the clamped-edge plate bending stress from the Roark flat-circular-plate closed form sigma_max = (3/4) * p * (r/t)^2, the required pane thickness that inverts that relation, the margin against a designer-supplied allowable stress, and the pane weight rollup over the window count, in pure Python, stdlib only. It pairs with fuselage-sizing (the barrel geometry around the aperture) and fuselage-skin-stringer (the skin and frame grid that surrounds the window, plus the skin cutout reinforcement context); this leaf owns the pane itself. It takes the ISA cabin and ambient pressures from isa-atmosphere (cross-cutting) and does not size the outflow or relief valves, the bulkheads, or the impact case.

Domain quick reference

• ISA pressure (paraphrase of the standard pressure formula owned at cross-cutting/units-atmos/isa-atmosphere): troposphere (0 to 11 km) P(h) = P0 * (1 - Lh/T0)^e with e = g0/(RL), and isothermal stratosphere (11 to 20 km) P(h) = P_tropo * exp(-g0*(h - 11 km) / (R*T_tropo)); valid 0 to 20000 m.
• Limit differential: the cabin ISA pressure minus the ambient ISA pressure at the flight altitude, dp_limit = P_cabin - P_ambient.
• Design differential: dp_design = dp_limit * certification_factor, with the default CERT_PRESSURE_FACTOR = 1.33 applied once as the certification pressure factor (the ultimate pressure check applies 1.33 times the normal operating differential pressure, a paraphrase of the FAR 25.365 cabin pressure rule, never a quote).
• Clamped-edge plate stress: sigma_max = (3/4) * p * (r/t)^2 in Pa (Roark flat-circular-plate case, clamped edge, uniform load). The maximum sits at the clamped edge and the 3/4 constant is independent of the Poisson ratio; the center stress 3*(1+nu)p(r/t)^2/8 is lower at nu = 0.33 (0.49875 versus 0.75 times p*(r/t)^2).
• Required pane thickness: t_req = r * sqrt((3/4) * p / sigma_allow), the exact inversion of the stress relation; pick the first standard gauge above it.
• Pane margin: margin = sigma_allow / sigma_computed - 1; a negative margin means the pane fails at the design differential.
• Pane weight: m = n * rho * pi * r^2 * t per rollup, a function of the pane volume only, so it takes no pressure argument.
• FAR-25 (14 CFR Part 25) sets the certification context for pressurized-cabin transport aeroplane structure; the closed-form plate result above is common conceptual sizing methodology.

Workflow

1. Collect the inputs: cabin altitude (m), flight altitude (m), pane radius r (m), the pane material allowable stress sigma_allow (Pa) and density rho (kg/m^3) chosen by the designer, the candidate gauge thickness t (m), and the window count.
2. Compute the design pressure differential traverse with design_pressure_differential(cabin_altitude_m, flight_altitude_m): the ISA cabin and ambient pressures at the two altitudes, the limit differential between them, and the design differential that applies the certification pressure factor.
3. Compute the clamped-circular-plate stress traverse with plate_max_stress_clamped_circular(pressure_pa, radius_m, thickness_m): the clamped-edge bending stress of the candidate gauge under the design differential.
4. Invert the stress relation for the required pane thickness with pane_thickness(pressure_pa, radius_m, allowable_stress_pa), the pane thickness inversion that returns the gauge running exactly at the allowable.
5. Check the pane margin with pane_margin(pressure_pa, radius_m, thickness_m, allowable_stress_pa) for the candidate gauges, and select the first standard gauge whose margin is positive at the design differential.
6. Roll up the pane weight with window_weight(radius_m, thickness_m, material_density_kg_m3, n_windows) for the selected gauge, giving the per-window mass and the total over the window count.
7. Run the gauge verification pass at the limit differential (the design differential without the certification factor) to confirm the design case governs and the selected gauge holds margin in both cases.
8. Confirm every result with the deterministic contract test scripts/test_window_aperture_sizing.py (step 8 confirmation).

Worked example

Cabin altitude 8000 ft (2438.40 m), flight altitude 12000 m, pane radius 0.15 m, acrylic pane (density 1190 kg/m^3) with a designer-supplied allowable of 50 MPa (the leaf hard-codes no material). Real module outputs:

• design_pressure_differential(2438.40, 12000.0): cabin pressure 75262.136558 Pa, ambient pressure 19330.062329 Pa, limit differential 55932.074230 Pa, design differential (times 1.33) 74389.658725 Pa, about 0.744 bar. The outflow-valve sibling quotes 75262 Pa for the same cabin, cross-consistent.
• plate_max_stress_clamped_circular at t = 6 mm: 34.870153 MPa, margin against 50 MPa 0.433891; at t = 5 mm: 50.213020 MPa, margin -0.004242.
• pane_thickness(74389.658725, 0.15, 50e6) = 0.005010640 m, so the 6 mm pane is the first standard gauge with positive margin.
• Limit-pressure (no certification factor) stress at t = 6 mm: 26.218160 MPa, margin 0.907075.
• window_weight(0.15, 0.006, 1190, 100): 0.504697 kg per window, 50.469686 kg over 100 windows.
• Sweep at r = 0.10 m: required thickness 3.340426 mm; at t = 10 mm the stress is 5.579224 MPa.

Verification

• Deterministic, offline checks: the contract test asserts the worked example anchors above (each pressure within 1 Pa, stresses within 1e-3 MPa, thickness within 1e-5 m, margins within 1e-4, weights within 1e-4 kg), the ISA anchors (sea level 101325 Pa, tropopause 22631.700910 Pa within 0.5 Pa), the scaling identities (stress linear in pressure, quadratic in radius, inverse quadratic in thickness; required thickness scales as the square root of the pressure and linearly with the radius; doubling the certification factor scales the required thickness by sqrt(2)), the round trip (the stress at the required thickness returns the allowable within 1e-6 relative), the clamped-edge-versus-center stress ordering at nu = 0.33, and the dict keys exactly as documented.
• Non-physical inputs raise ValueError: negative altitude or altitude above 20000 m, a flight altitude at or below the cabin altitude, a non-positive certification factor, and non-positive pressure/radius/thickness/allowable/density or a window count below

Pitfalls

• Leaving the small-deflection regime: the linear small-deflection plate stress is the conceptual sizing standard; a very thin, highly loaded pane can leave it (deflection of the order of the thickness), where membrane stiffening alters the load path. Keep the linear closed form and flag such layouts for a refined analysis.
• Sizing the pane on the limit differential alone: the certification pressure factor must be applied once to the limit differential to form the design differential that the gauge selection runs on; the limit-pressure pass is only the step 7 confirmation.
• Forgetting the clamped-edge location: the maximum clamped-edge stress sits at the rim, not the center; at nu = 0.33 the center stress is only 0.49875 versus 0.75 times p*(r/t)^2, so judging the pane on the center stress under-sizes it.
• Using a pane radius where the aperture radius belongs: r is the radius of the clamped pane, sized on the clear aperture plus the edge support; mixing the two under-sizes the pane.
• Applying the allowable without the material step: the leaf hard- codes no material, so the allowable stress and density are designer inputs; the weight rollup is density-driven and takes no pressure.
• Mixing units: pressures in Pa, lengths in m, stress in Pa, density in kg/m^3; a radius in mm with a thickness in m silently breaks the stress and thickness inversions.
• Passing zero or negative inputs; the module raises ValueError instead of returning a nonsense thickness or margin.

Related leaves

• skills/vehicle-design/sizing/fuselage-sizing (barrel geometry)
• skills/vehicle-design/sizing/cabin-outflow-valve-sizing (outflow)
• skills/vehicle-design/structures-integration/fuselage-skin-stringer (skin and frame grid around the aperture)
• skills/structures/fem/pressure-bulkhead (pressure dome structure)
• skills/structures/damage-tolerance/bird-strike (impact case)
• skills/cross-cutting/units-atmos/isa-atmosphere (pressure formula)

Behavior contract (gate 3)

The ISA pressures, design differential, clamped-edge plate stress, required pane thickness, margin and weight rollup relations are exercised by the gate 3 contract test: scripts/ test_window_aperture_sizing.py against scripts/ window_aperture_sizing_logic.py (stdlib unittest, offline). Run: python3 scripts/test_window_aperture_sizing.py

Compliance

• Standards referenced, not reproduced: FAR-25 is US government work (public domain); the clamped-circular-plate closed form is common structural methodology and the certification pressure factor is a paraphrase of the FAR 25.365 cabin pressure rule, summary-only per standards-map.yaml. No regulation text is quoted verbatim.
• compliance: STANDARDS-REF, gated: false.